Theorem 1stcclb 22503

Description: A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)

Hypothesis

Ref	Expression
1stcclb.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
1stcclb	⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐽,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem 1stcclb

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1stcclb.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	is1stc2 22501	. . 3 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))))
3	2	simprbi 496	. 2 ⊢ (𝐽 ∈ 1stω → ∀𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
4		eleq1 2826	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦))
5		eleq1 2826	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ 𝑧 ↔ 𝐴 ∈ 𝑧))
6	5	anbi1d 629	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → ((𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
7	6	rexbidv 3225	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))
8	4, 7	imbi12d 344	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
9	8	ralbidv 3120	. . . . 5 ⊢ (𝑤 = 𝐴 → (∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
10	9	anbi2d 628	. . . 4 ⊢ (𝑤 = 𝐴 → ((𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))) ↔ (𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))))
11	10	rexbidv 3225	. . 3 ⊢ (𝑤 = 𝐴 → (∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))) ↔ ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))))
12	11	rspcv 3547	. 2 ⊢ (𝐴 ∈ 𝑋 → (∀𝑤 ∈ 𝑋 ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝑤 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))))
13	3, 12	mpan9 506	1 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑧 ∈ 𝑥 (𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  ωcom 7687   ≼ cdom 8689  Topctop 21950  1^stωc1stc 22496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-pw 4532  df-uni 4837  df-1stc 22498

This theorem is referenced by:  1stcfb  22504  1stcrest  22512  lly1stc  22555  tx1stc  22709